426 



Dr. L. Silberstein on the Relation between 



so that P and P' have a symmetrical position within the 

 segment, as might have been expected. 



Thus i£ the semi-length o£ the gap * P'P or each of the 

 equal distances MP', MP, be denoted by n, we have the 

 surprisingly simple result 



Tan 77 = Tan 2 \. 



The connexion of these two characteristic points P, P r 

 with the Lobatchevskyan (arrowed) parallels through any 

 point Y is shown in fig. 6, which, after what was said before,, 

 scarcely calls for further explanations. 



Fig. 6. 



To resume : — 



Every Lobatchevskyan straight segment has intrinsically 

 associated with it a fair of characteristic points P and P', 

 symmetrically situated within it, ivhose mutual distance- 

 2rj is given by 



Tan ^= Tan 2 4, 



(12) 



and is thus a function only of the tbtal length 2\ of the 

 segment. 



It may be interesting to notice that for segments which are small 

 fractions of R we have, approximately. 77/11= (A/R) 2 , or 



PP' : OT = OT : 2#, (12a) 



that is to say, the " gap " is to the whole segment as this is to twice 

 the radius of curvature of the contemplated space. 



* This portion of the whole segment OUT deserves such a name 

 because if the end-point X of a vector "K=OX falls within it, there is 

 no negative vector —X with O as origin. Cf. Proj. Vector Algebra. 



